I think it's both confusing and wrong. I think there is a fundamental incompatibility, but I don't think the Atlantic article quite captured it.

At its most permissive, transgender individuals and advocates maintain that (1) there is a real, meaningful, substantive difference between being a man and a woman, but (2) which one you are (if either) is the one you say you are. That is, it is purely determined by internal desire. But it is real.

This is in tension with--if not outright conflict with--the "gender is socially constructed" idea that is both popular in some feminist circles, and is used as justification for permissive attitudes towards many of the different identities in LGBQIA+ (i.e. not T).

Indeed, solidarity within the transgender community and advocates thereof seem to have a considerable internal tension--I'm not sure how much it's expressed within the community in private as I'm not privy to that, but it's certainly papered over enthusiastically during any advocacy--between people who suffer from severe gender dysphoria ("man trapped in a woman's body" or "woman trapped in a man's body") and generally want maximal gender affirming surgery and hormonal therapy ("transmed") and care a great deal about "passing", and those who have a longing for, strong curiosity about, or transient but repeated feelings of wanting to be the opposite gender (but who are less likely to undergo full medical treatment and may be less concerned with passing; "tucute" — though fair warning, though I don’t know another term for it, I think this one is often considered a slur).

It's very difficult to reconcile these views with each other. You end up having to conclude that either (1) gender dysphoria is a sham and some people just have more vivid longings than others, (2) non-gender-dysphoric people aren't really trans, or (3) this phenomenon is actually way more complicated than anyone doing advocacy wants to let on, and practically every argument in favor of or against any sort of measures regarding trans people are bogus because they fail to recognize the profound complexity underlying the simple label.

Transphobes latch on to (1), as it would be really convenient for their outlook if it happened to be true, but I am pretty sure (3) is the reality.

Regarding "what is a woman", I think the question itself is problematic (along the lines that Azur Owle says). That is, when you have a near continuum of possibilities, even if you manage to construct a super-crisp definition that splits reality into two groups ("women" and "not-women"), if reality itself is not split that way, you have only managed to risk confusing yourself with the precision of your language, by making yourself think that cases just on either side of your arbitrary boundary are far more unalike that they are. For instance, you might decide that 6' is "tall" for a man. But if you were to do that and insist that a 5'11 3/4" man is very different from a 6'0 1/8" man but very much like a 5'7" man (in terms of height), you would be pretty confused.

When questions like these come up in the sciences, people mostly bail on the question of precise definition of a single term. Instead, they clarify what they're talking about by adding as many descriptors as necessary. Unfortunately, this doesn't work very well for trans people who do actually suffer from strong gender dysphoria because being labeled a "strongly gender dysphoric transgender woman" in every context is only likely to provoke stress and, in all but the very most tolerant communities, stereotype threat.

But if you want to actually answer the question "what is a woman--give me a perfect binary definition", the most rigorous answer that I know of is: "nuh-uh, not that simple".